Thèses sur le sujet « Gauge theorie »

Superconformal field theories have been widely explored over the last years, especially because of their large amount of symmetry which allows to derive exact results. The most prominent example within this class is the four-dimensional maximally supersymmetric N = 4 Super Yang-Mills theory, which played a pivotal role in the proposal of the AdS/CFT correspondence, a conjectured weak/strong duality relating superconformal gauge theories to string theories on curved backgrounds. Soon after the formulation of the correspondence the presence of integrable structures was discovered both in planar N = 4 SYM and in its string counterpart, and since then integrability in superconformal theories became one of the main research topics in theoretical physics. Interestingly, integrability in N = 4 was later shown to be intimately related to the presence of the so called dual conformal symmetry, a hidden symmetry of the planar amplitudes that puts even stronger constraints on their structure. This suggests that some crucial aspects of a theory can be investigated through the computation of its scattering amplitudes, which can thereby provide a powerful tool also for the study of less supersymmetric superconformal field theories. In particular it is essential to understand whether some of the beautiful properties of N = 4 survives when supersymmetry is not maximal. In this thesis we present computations of massless scattering amplitudes in two different not maximally supersymmetric conformal theories: N = 2 superconformal QCD in four dimensions and N = 6 Chern-Simons matter theory (ABJM) in three dimensions. In N = 2 SCQCD we compute all possible four-point amplitudes at one loop and the two-loop amplitude with fundamental fields as external legs. In ABJM we extend the two-loop computation of four-point scattering amplitudes and of the Sudakov form factor beyond the planar limit. We also discuss our results in relation to the corresponding ones in N = 4 SYM, paying particular attention to the possible presence of dual conformal invariance and to maximum transcendentality.

The thesis focuses on the study of various supersymmetric three-dimensional gauge theories, mainly with at least N = 3 supersymmetry. We range between very different theories and discuss several different aspects with the aim of validate our assumptions. Therefore, the leitmotiv of this work resides not so much in the topics we cover, but rather in the method that we use to obtain such results. This, in fact, consists in analysing the gauge invariant operators of the theory forming the so-called chiral ring. By having access to the chiral ring structure of the theory and to the operators forming it, we gain insight to the properties that needed to confirm or debunk our hypothesis. We will essentially use two different tools for counting and studying such chiral operators: the Hilbert series and the three-dimensional superconformal index. Thanks to the Hilbert series, we propose a quiver description for the mirror theories of the circle reduction of four-dimensional twisted χ(a2N) theories of class S. These mirrors are, in fact, described by "almost" star-shaped quivers containing both unitary and orthosymplectic gauge groups, along with hypermultiplets in the fundamental representation. On the other hand, by means of the superconformal index, we investigate the N = 2 preserving exactly marginal operators of the so called S-fold theories. In particular, we focus on two families of such theories, constructed by gauging the diagonal flavour symmetry of the T(U(N)) and T[2,12][2,12 ](SU(4)) theories. In addition, we also examine in detail the zero-form and one-form global symmetries of the Aharony-Bergman-Jafferis theories, with at least N = 6 supersymmetry, and with both orthosymplectic and unitary gauge groups. A number of dualities among all these theories are discovered and studied using the aforementioned tools.
The thesis focuses on the study of various supersymmetric three-dimensional gauge theories, mainly with at least N = 3 supersymmetry. We range between very different theories and discuss several different aspects with the aim of validate our assumptions. Therefore, the leitmotiv of this work resides not so much in the topics we cover, but rather in the method that we use to obtain such results. This, in fact, consists in analysing the gauge invariant operators of the theory forming the so-called chiral ring. By having access to the chiral ring structure of the theory and to the operators forming it, we gain insight to the properties that needed to confirm or debunk our hypothesis. We will essentially use two different tools for counting and studying such chiral operators: the Hilbert series and the three-dimensional superconformal index. Thanks to the Hilbert series, we propose a quiver description for the mirror theories of the circle reduction of four-dimensional twisted χ(a2N) theories of class S. These mirrors are, in fact, described by "almost" star-shaped quivers containing both unitary and orthosymplectic gauge groups, along with hypermultiplets in the fundamental representation. On the other hand, by means of the superconformal index, we investigate the N = 2 preserving exactly marginal operators of the so called S-fold theories. In particular, we focus on two families of such theories, constructed by gauging the diagonal flavour symmetry of the T(U(N)) and T[2,12][2,12 ](SU(4)) theories. In addition, we also examine in detail the zero-form and one-form global symmetries of the Aharony-Bergman-Jafferis theories, with at least N = 6 supersymmetry, and with both orthosymplectic and unitary gauge groups. A number of dualities among all these theories are discovered and studied using the aforementioned tools.

In questa tesi discutiamo un limite di particolari teorie di gauge 3d, le teorie T^sigma_rho[SU(N)], che ammetono una descrizione in termini di da quiver. Nel limite considerato, il numero di nodi è grande e i ranghi scalano quadraticamente con la lunghezza del quiver. Le energie libere sulla sfera e il topologically twisted index sono ottenuti usando la procedura della localizzazione supersimmetrica. Entrambi scalano quarticamente con la lunghezza del quiver e quadraticamente con N, con funzioni trilogaritmiche dipendenti dai dati del quiver come coefficienti. Le teorie precedentemente discusse con scaling N^2 \ln N sorgono come casi limite. Le SCFTs IR hanno duali di supergravità ben definiti in Tipo IIB: le energie libere corrispondono esattamente ai risultati olografici e gli indici, nel caso di un twist universale, riproducono correttamente l'entropia di un buco nero universale che può essere immerso nelle soluzioni olograficamente duali. Ogni teoria 3d descritta da un quiver bilanciato è legata ad una 5d, il cui modello matriciale è dominato dallo stesso punto di sella. Ciò conduce a strette relazioni tra osservabili BPS. In particolare, calcoliamo il valore di aspettazione dei Wilson loop in rappresentazioni antisimmetriche, trovando un perfetto accordo con il lato di gravità in un particolare esempio.
In this thesis we discuss a limit of 3d T^sigma _rho[SU(N)] quiver gauge theories in which the number of nodes is large and the ranks scale quadratically with the length of the quiver. The sphere free energies and topologically twisted indices are obtained using supersymmetric localization. Both scale quartically with the length of the quiver and quadratically with $N$, with trilogarithm functions depending on the quiver data as coefficients. Previously discussed theories with $N^2 \ln N$ scaling arise as limiting cases. The IR SCFTs have well-behaved supergravity duals in Type IIB: the free energies match precisely with holographic results and the indices, in case of a universal twist, correctly reproduce the entropy of an universal black hole which can be embedded in the holographically dual solutions. Each balanced 3d quiver theory is linked to a 5d parent, whose matrix model is related and dominated by the same saddle point, leading to close relations between BPS observables. In particular, we compute the expectation value of Wilson loops in antisymmetric representations, finding perfect agreement with the gravity side in a particular example.

Um das mikroskopische Verhalten der Gravitation zu beschreiben, ist es nötig, Quantenfeldtheorie und allgemeine Relativitätstheorie in einer vereinheitlichten Sprache zu formulieren. Eine Möglichkeit dieses Problem anzugehen ist es, die Punktteilchen der Quantenfeldtheorie durch fadenförmige Strings zu ersetzen. Allerdings erfordert die mathematische Konsistenz, dass sich die String in höherdimensionalen Raum-Zeiten bewegen; dies macht es jedoch sehr schwer, physikalische Konsequenzen zu extrahieren. Eine mögliche Lösung dieses Problems ist die Verwendung von String-Dualitäten, welche die Stringtheorie mittels holographischer Beschreibungen mit Eichtheorien auf dem Rand der Raum-Zeit verbinden. Die Dualitäten sind begründete Vermutungen, die die String- und Eichtheorie bei unterschiedlichen Werten der Kopplung gleichsetzen. Nicht zuletzt deshalb ist eine direkte Überprüfung der Dualitäten schwierig durchführbar. Hier hilft jedoch die sehr bemerkenswerte Tatsache, dass eine verborgene Eigenschaft der Vermutungen Integrabilität zu sein scheint, welche eine Extrapolation zwischen starker und schwacher Kopplung ermöglicht. Desweiteren kann das gesamte Spektrum, in gewissen vereinfachenden Grenzfällen, durch einen kompakten Satz von Bethe-Gleichungen ausgedrückt werden. Die Bethe-Gleichungen, welche aus Eichtheorierechnungen hergeleitet und geraten werden, bieten ein exzellentes Hilfsmittel, die vermuteten Dualitäten zu prüfen. Durch das Vergleichen der Vorhersagen der Gleichungen und expliziten Berechnungen in der Stringtheorie erhält man starke Argumente für die Gültigkeit der Vermutung und der angenommenen Integrabilität.
In this thesis we study superstring theory on AdS$_5\, \times\,$S$^5$, AdS$_3\,\times\,$S$^3$ and $\adsfour$. A shared feature of each theory is that their corresponding symmetry algebras allows for a decomposition under a $\mathbb{Z}_4$ grading. The grading can be realized through an automorphism which allows for a convenient construction of the string Lagrangians directly in terms of graded components. We adopt a uniform light-cone gauge and expand in a near plane wave limit, or equivalently, an expansion in transverse string coordinates. With a main focus on the two critical string theories, we perform a perturbative quantization up to quartic order in the number of fields. Each string theory is, through holographic descriptions, conjectured to be dual to lower dimensional gauge theories. The conjectures imply that the conformal dimensions of single trace operators in gauge theory should be equal to the energy of string states. What is more, through the use of integrable methods, one can write down a set of Bethe equations whose solutions encode the full spectral problem. One main theme of this thesis is to match the predictions of these equations, written in a language suitable for the light-cone gauge we employ, against explicit string theory calculations. We do this for a large class of string states and the perfect agreement we find lends strong support for the validity of the conjectures.

The goal of this thesis is to obtain and study new black hole solutions, both with and without supersymmetry, with a particular focus on multi-centered black holes in a cosmological background. After a review of matter-coupled N=2 gauged supergravity in four dimensions and of the classification of its supersymmetric solutions, a new supersymmetric black hole solution is obtained, which is the first with nontrivial running hyperscalars. Fake supergravity is a framework that allows to apply the methods used to classify supersymmetric solutions of supergravity also to theories without supersymmetry. A classification of fake supersymmetric solutions of a theory related to N=2, d=4 gauged supergravity is reviewed and used to construct dynamical solutions representing multiple black holes in a Friedmann-Lemaître-Robertson-Walker background. The physical properties of the single-centered case are then studied in some detail. More complicated solutions with rotation and NUT-charge, or with curved spatial slices, are obtained for a less general class of theories. Finally a recipe to obtain multi-centered black holes in an arbitrary FLRW universe and in arbitrary dimension is presented. These spacetimes are a multi-centered generalization of the charged McVittie black hole and are sourced by a U(1) gauge field and by a charged perfect fluid. As a particular subcase, these solutions describe an arbitrary number of black holes in a background that is locally anti-de Sitter space in cosmological coordinates. Some physical properties of the single-centered asymptotically AdS black hole are studied, showing in particular that a generalized first law of black hole dynamics is satisfied.

Cette thèse traite de plusieurs façons de construire une théorie quantiques des champs en quatre dimensions à partir de la théorie des cordes.

Dans une première partie nous étudions la construction d'une théorie Yang-Mills supersymétrique, couplée à un superchamp chiral dans la représentation adjointe, à partir de la théorie des cordes de type IIB sur une variété Calabi-Yau non compacte, avec des D-branes qui enroulent certaines sousvariétés. Les propriétés de
la théorie de jauge sont alors reflétées dans la structure
géométrique de la variété Calabi-Yau. En particulier, on peut calculer en principe le superpotentiel effectif de basse énergie qui décrit la structure des vides de la théorie de jauge en utilisant la théorie des cordes (topologiques). Malheureusement, en pratique, ceci n'est pas faisable. Il est remarquable qu'on puisse cependant montrer que la dynamique de basse énergie de la
théorie de jauge est codée par la géométrie d'une autre variété Calabi-Yau non compacte, reliée à la première par une transition géométrique. La théorie des cordes de type IIB sur cette deuxième variété, dans laquelle sont allumés des flux de fond appropriés, génère une théorie de jauge en quatre dimensions, qui n'est d'autre que la théorie effective de basse énergie de la théorie de
jauge originale. Ainsi, pour obtenir le superpotentiel effectif de basse énergie il suffit simplement de calculer certaines intégrales dans la deuxième géométrie Calabi-Yau, ce qui est faisable, au moins perturbativement. On trouve alors que le problème extrêmement difficile d'étudier la dynamique de basse
énergie d'une théorie de jauge non Abelienne a été réduit à celui de calculer certaines intégrales dans une géométrie connue. On peut démontrer que ces intégrales sont intimement reliées à certaines quantités dans un modèle de matrices holomorphes, et on peut alors calculer le superpotentiel effectif comme fonction de
certaines expressions du model de matrices. Il est remarquable que la série perturbative du modèle de matrices calcule alors le superpotentiel effectif non-perturbatif.

Ces relations étonnantes ont été découvertes et élaborée par plusieurs auteurs au cours des dernières années. Les résultats originaux de cette thèse comprennent la forme précise des relations de la ``géométrie spéciale" sur une variété Calabi-Yau
non compacte. Nous étudions en détail comment ces intégrales géométriques dépendent du cut-off, et leur relation à l'énergie libre du modèle de matrices. En particulier, sur une variété Calabi-Yau non compacte nous proposons une forme bilineaire sur le
produit direct de l'espace des formes avec l'espace des cycles, qui élimine toutes les divergences, sauf la divergence logarithmique. Notre analyse détaillée du modèle de matrices holomorphes clarifie aussi plusieurs aspects reliés à la méthode du col de ce modèle de matrices. Nous montrons en particulier qu'exiger une densité spectrale réelle restreint la forme de la
courbe Riemannienne qui apparaît dans la limite planaire du modèle de matrices. Çela nous donne des contraintes sur la forme du contour sur lequel les valeurs propres sont intégrées. Tous ces
résultats sont utilisés pour calculer explicitement l'énergie libre planaire d'un modèle de matrices avec un potentiel cubique.

La deuxième partie de cette thèse concerne la génération de théories de jauge supersymétriques en quatre dimensions comportant des aspects caractéristiques du modèle standard à partir de
compactifications de la supergravité en onze dimensions sur une variété G_2. Si cette variété contient une singularité conique, des fermions chiraux apparaissent dans la théorie de jauge en quatre dimensions ce qui conduit potentiellement à des anomalies. Nous montrons que, localement à chaque singularité, les anomalies
correspondantes sont annulées par une non-invariance de l'action classique au singularités (``anomaly inflow"). Malheureusement, aucune métrique d'une variété G_2 compacte n'est connue explicitement. Nous construisons ici des familles de métriques sur des variétés compactes faiblement G_2, qui contiennent deux singularités coniques. Les variétés faiblement G_2 ont des propriétés semblables aux propriétés des variétés G_2, et alors ces exemples explicites pourraient être utiles pour mieux comprendre la situation générique. Finalement, nous regardons la
relation entre la supergravité en onze dimensions et la théorie des cordes hétérotiques E_8\times E_8. Nous étudions en détail les anomalies qui apparaissent si la supergravité est formulée sur le produit d'un espace de dix dimensions et un intervalle. Encore une fois nous trouvons que les anomalies s'annulent localement sur
chaque bord de l'intervalle si on modifie l'action classique d'une façon appropriée.

The central theme in this thesis is compactifications: reductions of higher dimensional theories to lower dimensions and how the geometry of the compactification manifold determines features of the low energy physics. This is studied in the context of non-perturbative string theory in the framework of M-theory and F-theory. Supersymmetry requires the compactification manifold in F-theory to be an elliptically fibered Calabi-Yau, where the complex structure of the elliptic fibration is identified with the complexified coupling constant in type IIB string theory. The non-perturbative nature of the theory originates from the strong-weak duality of type IIB, which manifests itself as the SL(2;Z) modular transformation of the complex structure. Non-abelian gauge symmetries arise naturally in this framework and engineering Grand Uni ed Theories within F-theory has been an active area of research. Compactifications on Calabi-Yau four-folds give rise to gauge theories with N = 1 supersymmetry in four dimensions coupled to gravity. In the first part of this thesis we focus on abelian gauge symmetries in F-theory, which are essential in SU(5) GUTs for forbidding couplings which result in fast proton decay. These arise from rational sections in the elliptic fibration and from the geometric constraints on these sections one can determine the set of possible U(1) charges of GUT matter representations. Armed with this constrained set of charges we then proceed to study the phenomenology of these abelian gauge symmetries in the context of SU(5) GUT models. We analyse their e ectiveness at suppressing proton decay operators and explore the types of realistic flavour textures that can be generated using the Froggatt-Nielsen mechanism. In the latter part of this thesis the focal point changes to M5-branes, one of the two fundamental objects of M-theory. The theory of multiple M5-branes is known to be a 6d N = (2; 0) superconformal eld theory, of which only the space-time symmetries and abelian equations of motion have been determined. In spite of this, fascinating correspondences have been shown to arise from the reduction of the M5-brane theory to lower dimensions. In particular, supersymmetric observables in the reduced theories capture non-trivial aspects of the geometry of the compactification manifold. The final chapter of this thesis studies the compactification of the 6d N = (2; 0) theory on the two-sphere as a step towards deriving a correspondence related to four-manifolds.

A class of diffeomorphism invariant gauge theories is studied. The action for this class of theories can be formulated as a generalisation of the well known topological BF-theories with a potential for the B-field or in a pure connection formulation. When the gauge group is chosen to be SU(2) the theory describes gravity. For a larger gauge group G one gets a unified model of gravity and Yang-Mills fields. A background for the theory is chosen which breaks the gauge group G by selecting in it a preferred SU(2) subgroup which describes the gravitational sector. The Yang-Mills sector is described by the part of the gauge group that commutes with this SU(2). Thus, when the action is expanded around this background the spectrum of the linearised theory consists of the usual gravitons plus Yang-Mills fields. In addition, there is a set of massive scalar fields that are charged both under the gravitational and Yang-Mills subgroups. The latter sector is described by the part of the gauge group that does not commute with SU(2). A fermionic Lagrangian is also proposed which can be coupled to the BF plus potential formulation.

Thesis (Ph. D.)--University of California, San Diego, 2007.
Title from first page of PDF file (viewed June 26, 2007). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 154-160).

Dimensional regularisation is formulated without using the assumption that f d(^D)k(k(^2))(^n) = 0. Alternative definitions of ϵ(_kλµv) and γ(^5) are also considered. In the reformulated scheme, quadratic divergences are present, in general, in the scalar and gauge boson self-energies, and remain unregularised. The possible cancellation of such divergences is investigated. Phenomenological aspects of unified gauge theories are studied.

Includes bibliographical references.
The thesis is organized as follows. Part I is a general introduction to LGT. The theory is discussed from first principles, so that for the interested reader no previous knowledge is required, although it is assumed that he/she will be familiar with the rudiments of relativistic quantum mechanics. Part II is a review of QCD on the lattice at finite temperature and density. Monte Carlo results and analytical methods are discussed. An attempt has been made to include most relevant data up to the end of 1987, and to update some earlier reviews existing on the subject. To facilitate an understanding of the techniques used in LGT, provision has been made in the form of a separate Chapter on Group Theory and Integration, as well as four Appendices, one of which deals with Grassmann variables and integration.

Die vorliegende Dissertationarbeit ist dem Problem der analytischen Beschreibung des Confinement-Mechanismus in der QCD und in anderen Eichtheorien gewidment. Als Leitprinzip der Arbeit wurde das sogenannte Wilsonsche-Confinement-Kriterium gewählt, gemäss welchem diese Erscheinung durch eine effektive Stringtheorie beschrieben werden kann. Die entstehenden Strings des Eichfeldes verbinden farbige-Objekte (Quarks, Gluonen) miteinander und hindern ihr Auseinandergehen auf makroskopische Abstände. Es werden verschiedene Verfahren der Ableitung dieser Stringstheorien aus unterschiedlichen Eichtheorien, einschliesslich der QCD, vorgestellt. Kapitel 2 enthält die Untersuchung der nichtlokalen effektiven Stringwirkung, die im Rahmen des sogenannten stochastischen Vakuum-Modells der QCD entsteht, wobei die Wechselwirkung zwischen den Elementen der String-Weltfläche durch den phänomenologischen Background-Gluon-Propagator vermittelt wird. Durch Entwicklung dieser Wirkung nach Ableitungen wurden die ersten Terme niedrigster Ordnung bestimmt. Die ersten beiden Terme dieser Entwicklung sind die Nambu-Goto- und Rigidity-Terme mit Kopplungskonstanten, die sich durch das Gluon-Kondensat und die Korrelationlänge des QCD-Vakuums ausdrücken lassen. Die Vorzeichen dieser Konstanten zeigen, dass die durch dieses Verfahren erhaltenen Strings stabil sind. Danach wurde eine mögliche Lösung des ``Crumpling'' Problems auf der Basis eines zusätzlichen topologischen Stringtermes im Instantongas-Modell des QCD-Vakuums vorgestellt. Mittels Störungstheorie im nicht-störungstheoretischen QCD-Background berechneten wir zusätzliche-Korrekturen zur ursprünglichen nicht-störungstheoretischen Stringwirkung. Diese Korrekturen führen zu neuen Formen der nichtlokalen effektiven Stringwirkung, die den störungstheoretischen Gluon-Propagator im Backgroundfeld zwischen den Elementen der Weltfläche enthalten. Durch Ableitungsentwicklung dieser Wirkung bekommen wir eine Korrektur zur Kopplungskonstante des Rigidity-Terms; die Stringsspannung des Nambu-Goto-Terms jedoch bleibt unverändert. Am Ende dieses Kapitels wurde der Hamilton-Operator des QCD-Strings mit spinlosen Quarks hergeleitet, der der effektiven Stringwirkung mit Rigidity-Term entspricht. Dieser Hamilton-Operator liefert einen Korrekturterm zur Wechselwirkung im relativistischen Quarkmodell-Operator. Im Kapitel 3 untersuchten wir das Problem der Stringdarstellung von Abelsch-projezierten Eichtheorien. Als erstes wurde die Herleitung der Stringdarstellung der erzeugenden Funktion für das einfachste Modell dieser Art, d.h. die Abelsch-projezierte SU(2)-QCD gegeben, die einem dualen Abelschen Higgs-Modell mit äusseren elektrisch geladendenen Teilchen äquivalent ist. Der Vorteil dieses Stringszuganges im Vergleich zum Zugang des stochastischen Vakuum-Modells der QCD besteht in der Berücksichtigung der Integration über String-Weltflächen, die auf Grund der Integration über den Singulärteil der Higgsfeld-Phase entsteht. Zusätzlich zur Stringdarstellung der erzeugenden Funktion wurde im London-Limes die Stringdarstellung für die erzeugenden Funktionale der Feldstärke- und Monopolstromkorrelatoren hergeleitet. Dies gab uns die Möglichkeit, die entsprechenden bilokalen Kumulanten zu finden und zu zeigen, dass die bilokalen Kumulanten der Feldstärke für grosse Abstände das gleiche Verhalten wie die entsprechenden eichinvarianten Kumulanten der QCD zeigen. Das Letztere wurde durch das stochastische Vakuum-Modell vorhergesagt und durch Gitterexperimente berechnet. Dieses Ergebnis unterstützt einerseits die Methode der Abelschen Projektion und gibt anderseits dem stochastischen Vakuum-Modell der QCD einen neuen feldtheoretischen Status. Danach erweiterten wir unsere Analyse über den Rahmen des London-Limes hinaus untersuchten den Zusammenhang von quartischen Kumulanten und bilokalen Kumulanten. Anschliessend wurde die Stringdarstellung der SU(3)-Gluodynamik hergeleitet. Dabei wurde die Stringdarstellung für ein entsprechendes duales Modell formuliert, das drei Arten des magnetischen Higgs-Feldes enthält. Infolgedessen liefert das Modell drei Strings, von denen nur zwei wirklich unabhängig sind. Alle diese Strings wechselwirken untereinander durch Austausch zweier massiver dualer Eichbosonen. Ausserdem erhielten wir die bilokalen Kumulanten des effektiven dualen Modells der SU(3)-Gluodynamik. Die entsprechenden bilokalen Kumulanten zeigen für grosse Abstände ein Verhalten wie es durch das stochastische Vakuum-Modell vorhergesagt wurde. Zum Schluss dieses Kapitels geben wir eine nützliche Darstellung für erzeugende Funktionen von Abelsch-projezierten Theorien in Form von Integralen über Monopolströme an. Im Kapitel 4 wurde ein weiteres Modell untersucht, das eine analytische Beschreibung des Confinement-Mechanismus zulässt, nämlich die 3D kompakte QED. Für den Wilson-Loop der entsprechenden Theorie mit Monopoldichten wurde die Äquivalenz zur sogenannten Confining-Stringtheorie demonstriert. Ausserdem wurde das Verhalten der bilokalen Kumulante der Feldstärke im Limes schwacher Felder untersucht. Dieses Verhalten befindet sich ebenfalls in Übereinstimmung mit den Voraussagen des stochastischen Vakuum-Modells. Erwartungsgemäss sind die Stringdarstellungen der erzeugenden Funktionen der 3D kompakten QED im Limes schwacher Felder und der dualen Abelschen Higgs-Modelle sehr ähnlich. Wir zeigten ausserdem, dass diese Entsprechung nicht zufällig ist. Die 3D kompakte QED ergibt sich nämlich im Limes verschwindender Eichbosonmasse aus dem 3D Abelschen Higgs-Modell mit äusseren Monopolen. Zum Schluss wurde ein allgemeines Verfahren der Beschreibung der Anregungen der Stringweltfläche in Abelsch-projezierten Theorien (kompakte QED und QCD) ausgearbeitet. Es ist auf der Methode der nicht-linearen Sigma-Modelle gegründet und gibt eine Möglichkeit, die in diesen Fluktuationen quadratische Effektive Wirkung zu erhalten. In der Dissertation wurden analytische nicht-störungstheoretische Verfahren ausgearbeitet, die neue Informationen über den Confinement-Mechanismus in der QCD und anderen Eichtheorien liefern und zum besseren Verständnis der Vakuumstruktur dieser Theorien beitragen können. Sie sind insbesondere relevant für die Herleitung effektiver Stringtheorien aus Eichtheorien.
The main problem addressed in the present Dissertation was an attempt of an analytical description of confinement in QCD and other gauge theories. As a guiding principle for our investigations served the so-called Wilson's picture of confinement, according to which this phenomenon can be described in terms of some effective theory of strings, joining coloured objects to each other and preventing them from moving apart to macroscopic distances. In this Dissertation, we have proceeded with a derivation of such string theories corresponding to various gauge ones, including QCD, i.e. with the solution of the problem of string representation of gauge theories. We have started our analysis with the nonlocal string effective action, arising within the so-called Stochastic Vacuum Model of QCD, where the interaction between the string world-sheet elements is mediated by the phenomenological background gluon propagator. By performing the derivative expansion of this action, we have derived the first few terms of a string Lagrangian. The first two nontrivial of them turned out to be the Nambu-Goto and rigidity terms with the coupling constants expressed completely via the gluonic condensate and correlation length of the QCD vacuum. The signs of these constants ensure the stability of strings in the so-obtained effective string theory. After that, we have investigated the problem of crumpling for the string world-sheets by derivation of the topological string term in the instanton gas model of the gluodynamics vacuum. Next, by making use of perturbation theory in the nonperturbative QCD vacuum, we have calculated perturbative corrections to the obtained string effective action. Those lead to a new form of the nonlocal string effective action with the propagator between the elements of the world-sheet being the one of a perturbative gluon in the confining background. By the derivative expansion of this action, we got a correction to the rigidity term coupling constant, whereas the string tension of the Nambu-Goto term occurs to get no corrections due to perturbative gluonic exchanges. Finally, we have derived the Hamiltonian of QCD string with spinless quarks at the ends, associated with the obtained string effective action including the rigidity term. In the particular case of vanishing orbital momentum of the system, this Hamiltonian reduces to that of the so-called relativistic quark model, albeit with some modifications due to the rigidity term, which might have some influence on the dynamics of the QCD string with quarks. All these topics have been elaborated on in Section 2, and form the essence of the string representation of QCD within the Stochastic Vacuum Model. In Section 3, we have addressed the problem of string representation of Abelian-projected theories. In this way, we have started with the string representation for the partition function of the simplest model of this kind, namely the Abelian-projected SU(2)-QCD, which is argued to be the dual Abelian Higgs Model with external electrically charged particles. The advantage of this approach to the string representation of QCD w.r.t. the one based on the Stochastic Vacuum Model is a possibility to get an integration over the string world-sheets, resulting from the integration over the singular part of the phase of the Higgs field. After the string representation of the partition function in the London limit, we have proceeded with the string representation for the generating functionals of the field strength and monopole current correlators. Those enabled us to find the corresponding bilocal cumulants and demonstrate that the large-distance asymptotic behaviour of the bilocal field strength cumulant matches the one of the corresponding gauge-invariant cumulant in QCD, predicted by the Stochastic Vacuum Model and measured in the lattice experiments. This result supports the method of Abelian projection on the one hand and gives a new field-theoretical status to the Stochastic Vacuum Model on the other hand. After that, we have extended our analysis beyond the London limit, and studied the relation of the quartic cumulant, which appears there, with the bilocal one in the London limit. Next, by making use of the Abelian projection method, we have addressed the problem of string representation of the SU(3)-gluodynamics. Namely, we have casted the related dual model, containing three types of magnetic Higgs fields, into the string form. Consequently, the latter one turned out to contain three types of strings, among which only two ones were actually independent. As a result, we have found, that both the ensemble of strings as a whole and individual strings display confining properties in a sense that all types of strings (self)interact via the exchanges of the massive dual gauge bosons. We have also derived bilocal cumulants in the effective dual model of confinement, corresponding to the SU(3)-gluodynamics, and they turned out to be also in line with the ones predicted by the Stochastic Vacuum Model. In conclusion of this topic, we have obtained another useful representation for the partition functions of the Abelian-projected theories in the form of an integral over the monopole currents. In Section 4, we have studied another model, allowing for an analytical description of confinement, which is 3D compact QED. In this way, by virtue of the integral over the monopole densities, we have derived string representation for the Wilson loop in this theory and demonstrated the correspondence of this representation to another recently found one, the so-called confining string theory. After that, we have calculated the bilocal cumulant of the field strength tensors in the weak-field limit of the model under study. It also turned out to be in line with the general concepts of the Stochastic Vacuum Model and therefore matches the corresponding results known from the lattice measurements in QCD and found analytically for the effective Abelian-projected theories in the previous Section. Besides that, string representations of the partition functions of the weak-field limit of 3D compact QED and of the dual Abelian Higgs Model turned out to be also quite similar. We have illustrated later on that this correspondence is not accidental. Namely, we have shown that 3D compact QED is nothing else, but the limiting case of 3D Abelian Higgs Model with external monopoles, corresponding to the vanishing gauge boson mass. Finally, we have elaborated on a unified method of description of the string world-sheet excitations in the Abelian-projected theories, compact QED, and QCD, based on the techniques of nonlinear sigma-models, and obtained the effective action, quadratic in the world-sheet fluctuations. In conclusion, the proposed nonperturbative techniques provide us with some new information on the mechanisms of confinement in QCD and other gauge theories and shed some light on the vacuum structure of these theories. They also show the relevance of string theory to the description of this phenomenon and yield several prescriptions for the construction of the adequate string theories from the corresponding gauge ones.

Quiver gauge theories are widely studied in the context of AdS/CFT, which establishes a correspondence between CFTs and string theories. CFTs in turn offer a map between quantum states and Gauge Invariant Operators (GIOs). This thesis presents results on the counting and correlators of holomorphic GIOs in quiver gauge theories with flavour symmetries, in the zero coupling limit. We first give a prescription to build a basis of holomorphic matrix invariants, labelled by representation theory data. A fi nite N counting function of these GIOs is then given in terms of Littlewood-Richardson coefficients. In the large N limit, the generating function simpli fies to an in finite product of determinants, which depend only on the weighted adjacency matrix associated with the quiver. The building block of this product has a counting interpretation by itself, expressed in terms of words formed by partially commuting letters associated with closed loops in the quiver. This is a new relation between counting problems in gauge theory and the Cartier-Foata monoid. We compute the free fi eld two and three point functions of the matrix invariants. These have a non-trivial dependence on the structure of the operators and on the ranks of the gauge and flavour symmetries: our results are exact in the ranks, and their expansions contain information beyond the planar limit. We introduce a class of permutation centraliser algebras, which give a precise characterisation of the minimal set of charges needed to distinguish arbitrary matrix invariants. For the two-matrix model, the relevant non-commutative algebra is parametrised by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators. The structure of the algebra, notably its dimension, its centre and its maximally commuting sub-algebra, is related to Littlewood-Richardson numbers for composing Young diagrams.

Deformations of the original F-theory background are proposed. These lead to multiple new dualities and physical phenomena. We concentrate on one model where we let seven-branes wrap a multi-centered Taub-NUT space instead of R4. This configuration provides a successful F-theory embedding of a class of recently proposed four-dimensional N = 2 superconformal (SCFT) à la Gaiotto. Aspects of Argyres- Seiberg duality, of the new Gaiotto duality, as well as of the branes network of Benini- Benvenuti and Tachikawa are captured by our construction. The supergravity theory for the conformal case is also briefly discussed. Extending our construction to the non-conformal case, we find interesting cascading behavior in four-dimensional gauge theories with N = 2 supersymmetry. Since the analysis of this unexpected phenomenon is quite difficult in the language of type IIB/F-theory, we turn to the type IIA/M-theory description where the origin of the N = 2 cascade is clarified. Using the T-dual type IIA brane language, we first start by studying the N = 1 supersymmetric cascading gauge theory found in type IIB string theory on p regular and M fractional D3-branes at the tip of the conifold. We reproduce the supersymmetric vacuum struc- ture of this theory. We also show that the IIA analog of the non-supersymmetric state found by Kachru, Pearson and Verlinde in the IIB description is metastable in string theory, but the barrier for tunneling to the supersymmetric vacuum goes to infinity in the field theory limit. We then use the techniques we have developed to analyze the N = 2 supersymmetric gauge theory corresponding to regular and fractional D3-branes on a near-singular K3, and clarify the origin of the cascade in this theory.
Différentes déformations de la géométrie originale de la théorie F sont proposées. Ces dernières génèrent une multitude de nouvelles dualités ainsi que de nouveaux phénomènes physiques. Nous nous concentrons sur un seul modèle où les membranes en sept dimensions spatiales s'enveloppent autour d'un espace Taub-NUT avec multi-centres au lieux de l'espace R4 original. Cette configuration génère avec succès la réalisation, en théorie F, d'une famille de théories de jauges superconformes en quatres dimensions avec N = 2 supersymétries nouvellement proposées par Gaiotto. Deplus, plusieurs aspects de la dualité d'Argyres-Seiberg, de la nouvelle dualité de Gaiotto ainsi que du réseaux de membranes de Benini-Benvenuti et Tachikawa sont réalisés par notre construction. La théorie de supergravité pour le cas conforme est brièvement discutée. La généralisation de notre construction au cas non-conforme mène à l'observation surprenante de cascade chez les théories de jauges avec N = 2 supersymétries en quatres dimensions. Puisque l'analyse de ce phénomène est difficile dans le language de type IIB/ théorie F, nous nous tournons vers le type IIA/theorie M où l'origine de ce phénomène est élucidée. En utilisant le langage des membranes en type IIA sous la dualité-T, nous débutons par l'étude de cascade chez les théories de jauges avec N = 1 supersymétrie tel que présenté en type IIB avec p membranes D3 régulières et M membranes D3 fractionnaires situées au bout d'un espace conifold. Nous reproduisons avec succès la structure du vide supersymétrique de cette théorie. Aussi, nous démontrons que l'analogue en type IIA des états non-supersymmetriques découverts par Kachru, Pearson et Verlinde en type IIB sont métastables en théorie des cordes alors que la barrière permettant de passer au vide supersymmetrique tant vers l'infinie dans la limite de la théorie des champs. Nous utilisons finalement les techniques que nous avons développées afin d'analyser la théorie de jauge supersymmétrique avec N = 2 correspondante à des membranes D3 régulières et fractionnaires sur un espace K3 presque singulier et clarifions l'origine du mécanisme de cascade dans cette théorie.

Thesis (Ph. D.)--University of California, San Diego, 2006.
Title from first page of PDF file (viewed June 27, 2006). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 159-165).

Various effects of finite fermionic densities in gauge theories are studied. The phase structure of SU(N) gauge theories with fermions as a function of imaginary chemical potential is related to the confining properties of the theory. This phase structure is controlled by a remnant of the Z[sub N] symmetry which is present in the absence of fermions. At high temperature the theory has a first-order phase transition as a function of imaginary chemical potential. This transition is expected to be absent in the low-temperature phase. It is shown that properties of the theory at nonzero fermion density can be deduced from its behaviour at finite imaginary chemical potential. Anomalies in gauge theories are introduced using various two-dimensional models. In particular, the chiral Schwinger model is shown to be consistent despite being anomalous. The effects of finite densities in anomalous gauge theories is investigated. It is found that, contrary to some recent claims, an effective Harniltonian (obtained by integrating out the fermions) cannot be obtained by the simple inclusion of a Chern-Simons term multiplying the fermionic chemical potential. The importance of dynamical effects is stressed and a mechanism for producing primordial magnetic field is suggested.
Science, Faculty of
Physics and Astronomy, Department of
Graduate

Lattice gauge theories, through Monte-Carlo simulations, provide the most powerful methods available for the non-perturbative study of many models. These techniques, however, become very inefficient as we approach the continuum limit, a problem known as Critical Slowing Down. Over recent years cluster methods have generated significant improvements over established techniques. In part I of this thesis we introduce such an algorithm for the Z₂ Kalb-Ramond model in four dimensions, and find that we can improve the efficiency of the simulation by orders of magnitude. In the second part of this thesis we make preliminary investigations towards using duality transformations as an aid to improving numerical simulations, (where by duality we mean an extension of the technique of Kramers and Wannier). We study the compact U(1) lattice gauge theory in (2+1) dimensions as an example. The dual to this model is known to be a discrete Gaussian Solid On Solid model. We find the discrete nature of the model makes each update faster. However the structures which develop at high temperature, make naive simulations, overall, inefficient.

This thesis is concerned with so called chiral gauge theories, also known as self dual gauge theories. In particular, the aim of this thesis to investigate the role that chiral gauge theories play in duality symmetries in lower dimensions through dimensional reduction. Chapter one serves as an introduction to the notions of duality in field and string theory. The problems of formulating well defined actions for self-dual gauge theories are introduced as well as a brief presentation of the different approaches used to over come these problems. Chapter two introduces dimensional reduction and demonstrates how duality symmetries arise from the dimensional reduction of self-dual theories in a variety of dimensions and on different compact spaces. Examples are presented where the couplings of the resulting theories are calculated explicitly in terms of the geometrical data of the compact space. The duality generators acting on these couplings are also calculated explicitly and related to the geometry/topology of the compact space. Chapter three deals with the idea of duality manifest actions and their relation to the self-dual theories in higher dimensions. Non-linear Born-Infeld type actions are introduced and again dimensional reduction is shown to play a role in the duality of the Born-Infeld action. This leads to a duality manifest version of the Born-Infeld action. Chapter four describes perhaps the main application of this thesis. The effective action of the M-theory five brane wrapped on a torus is identified with the effective action of the IIB D-3 brane dimensionally reduced on a circle (after some appropriate world volume dualizations). The IIB S-duality then arises as a result of the modular symmetry of the torus. The final chapter contains a brief summary and a hint of further directions for research that were outside the scope of this thesis.

There has been great progress in recent years in the understanding of the mathematical structure of scattering amplitudes in Quantum Field Theory as well as the development of powerful methods for their calculation, particularly in the arena of N = 4 Super Yang-Mills where hidden and manifest symmetries lead to striking simplifications. In this thesis, we will discuss the extensions of such methods away from the case of on-shell amplitudes in conformal N = 4. After introducing the necessary mathematical background and physical setting, we consider in Chapter Three the form factors of BPS operators in N = 4 Super Yang- Mills. These objects have several physical applications, and share many properties with scattering amplitudes. However, they are off-shell, which makes them a natural starting point to set out in the direction of correlation functions. After demonstrating the computation of form factors by BCFW recursion and unitarity based methods, we go on to show how the scalar form factor can be supersymmetrised to encompass the full stress-tensor multiplet. In Chapter Four, we discuss the Sudakov form factor in ABJM Theory. This object, which first appears at two loops and controls the IR divergences of the theory, is computed by generalised unitarity. In particular, we note that the maximal transcendentality of three dimensional integrals is related to particular triple cuts. Finally, in Chapter Five we consider massive amplitudes on the Coulomb Branch of N = 4 at one loop. Here we find that vertex cut conditions inherited from the embedding of the theory in String Theory lead to a restricted class of massive integrals.

In this thesis we study partition functions of supersymmetric gauge theories on compact backgrounds in various dimensions, with particular focus on infinite dimensional symmetry algebras encoded in these observables. The compact space partition functions of the considered theories can be decomposed into products of holomorphic blocks which are identified with partition functions on elementary geometries. Partition functions on different compact spaces can be obtained by fusing the holomorphic blocks with pairings reflecting the geometric decomposition of the backgrounds. An example of this phenomenon is given by the S4 partition function of 4d N = 2 theories, which can be written as an integral of two copies of the R4 Nekrasov partition function. Remarkably, the AGT correspondence identifies the S4 partition function of class S theories with Liouville CFT correlators. The perturbative integrand is identified with the product of CFT 3-point functions, while each copy of the non-perturbative instanton partition function is identified with conformal blocks of the Virasoro algebra. In this work we define a class of q-deformed CFT correlators, where chiral blocks are controlled by the q-Virasoro algebra and are identified with R4xS1 instanton partition functions. We derive the 3-point functions for two different q-deformed CFTs, and we show that non-chiral correlators can be identified with S5 and S4xS1 partition functions of certain 5d N = 1 theories. Moreover, particular degenerate correlators are mapped to S3 and S2xS1 partition functions of 3d N = 2 theories. This fits the interpretation of the 3d theories as codimension two defects. We also study 4d N = 1 theories on T2 fibrations over S2. We prove that when anomalies are canceled, the compact space partition functions can be expressed through holomorphic blocks associated to R2xT 2. We argue that for particular theories these objects descend from R4xT 2 partition functions, which we identify with the chiral blocks of an elliptically deformed Virasoro algebra.

In this thesis we discuss supersymmetric gauge theories, focusing on exact results achieved using methods of integrability. For the larger portion of the thesis we study the N = 4 super Yang-Mills theory in the planar limit, a recurring topic being the Konishi anomalous dimension, which is roughly the analogue for the mass of the proton in quantum chromodynamics. The N = 4 supersymmetric Yang-Mills theory is known to be integrable in the planar limit, which opens up a wealth of techniques one can employ in order to nd results in this limit valid at any value of the coupling. We begin with perturbation theory where the integrability of the theory rst manifests itself. Here we showcase the rst exact result, the so-called slope function, which is the linear small spin expansion coecient of the generalized Konishi anomalous dimension. We then move on to exact results mainly achieved using the novel quantum spectral curve approach, the method allowing one to nd scaling dimensions of operators at arbitrary values of the coupling. As an example we nd the second coecient in the small spin expansion after the slope, which we call the curvature function. This allows us to extract non-trivial information about the Konishi operator. Methods of integrability are also applicable to other supersymmetric gauge theories such as ABJM, which in fact shares many similarities with N = 4 super Yang-Mills. We briefly review these parallel developments in the last chapter of the thesis.

This work presents recent developments on brane tilings and their vacuum moduli spaces. Brane tilings are bipartite periodic graphs on the torus and represent 4d N = 1 supersymmetric worldvolume theories living on D3-branes probing Calabi-Yau 3-fold singularities. The graph and combinatorial properties of brane tilings make the set of supersymmetric quiver theories represented by them one of the largest and richest known so far. The aim of this work is to give a concise pedagogical introduction to brane tilings and a summary on recent exciting advancement on their classification, dualities and construction. At first, particular focus is given on counting distinct Abelian orbifolds of the form C3/[gamma]. The presented counting of Abelian orbifolds of C3 and in more general of CD gives a first insight on the rich combinatorial nature of brane tilings. Following the classification theme, the work proceeds with the identification of all brane tilings whose mesonic moduli spaces as toric Calabi-Yau 3-folds are represented by reflexive polygons. There are 16 of these special convex lattice polygons. It is shown that 30 brane tilings are associated with them. Some of these brane tilings are related by a correspondence known as toric duality. The classification of brane tilings with reflexive toric diagrams led to the discovery of a new correspondence between brane tilings which we call specular duality. The new correspondence identifies brane tilings with the same master space - the combined mesonic and baryonic moduli space. As a by-product, the new correspondence paves the way for constructing brane tilings which are not confined to the torus but are on Riemann surfaces with arbitrary genus. We give the first classification of genus 2 brane tilings, illustrate the corresponding supersymmetric quiver theories and analyse their vacuum moduli spaces.

It is expected that the number of flavors in a gauge theory plays an important role in model building for physics beyond the standard model. We study the phase structure of the 12 flavor case through lattice simulations using a Rational Hybrid Monte Carlo (RHMC) algorithm for different masses, betas, and volumes, to investigate the question of conformality for this number of flavors. In particular, we analyze the Fisher's zeroes, in the vicinity of the endpoint of a line of first order phase transitions. This is motivated by previous studies that show how the complex renormalization group (RG) flows can be understood by looking at the zeros. The pinching of the imaginary part of these zeros with respect to increasing volume provides information about a possible unconventional continuum limit. We also study the mass spectrum of a multiflavor linear sigma model with a splitting of fermion masses. The single mass linear sigma model successfully described a light sigma in accordance to recent lattice results. The extension to two masses predicts an unusual ordering of scalar masses, providing incentive for further lattice simulations with split quark mass.

Taking advantage of the integrable structures emergent in the theory, non-local observables such as null polygonal Wilson loops are studied in 4d planar N = 4 Super Yang-Mills. Their duality with the 4d gluon scattering amplitudes makes the analysis even more interesting. The so-called Pentagon approach, an application of the Operator Product Expansion (OPE) method to the null polygonal Wilsol loops, makes possible a non-perturbative evaluation of these objects. They are recast as an OPE series over the 2d GKP flux-tube excitations, a description reminescent of the QCD flux-tube stretching between quarks. The integrability of the flux-tube allows us to evaluate the series, in principle, for any value of the coupling constant. From this analysis, several results have been obtained. In the strong coupling regime we reproduced the TBA-like equations expected from the minimal area problem in string theory, finding agreement with the AdS/CFT prediction. In this respect, of fundamental importance is the emergence of effective bound states between elementary fermionic excitations. Along the way, we encountered some intriguing analogies between these null polygonal Wilson loops and the Nekrasov instanton partition function Z for N = 2 theories. Furthermore, a new non-perturbative enhancement of the classical string argument has been confirmed, stemming from the dynamics of the string in the five sphere S5 and described by the non-linear σ-model O(6). Some properties of a fundamental building block of the OPE series, the SU(4) structure of the form factors of a specific twist operator P, have been analysed. This SU(4) matrix part is given a representation in terms of rational functions, organized in a Young tableaux pattern.

This thesis is dedicated to the study of Wilson loops in supersymmetric gauge theories, we will study their properties and how they are related to physical observables, such as the Bremsstrahlung function that controls to the energy radiated by an accelerating particle. We will give various examples of the computation of their expectations values using perturbative and non-perturbative techniques. For SU(N) superconformal QCD we perform a three-loop calculation of the generalized cusp anomalous dimension of the BPS Wilson loop operator using HQET formalism. We obtain an expression which is valid at generic geometric and internal angles and finite gauge group rank N. For equal and opposite angles this expression vanishes, proving that at these points the cusp becomes BPS. From its small angle expansion we derive the corresponding Bremsstrahlung function at three loops, matching the matrix model prediction given in terms of derivatives of the Wilson loop on the ellipsoid.
This thesis is dedicated to the study of Wilson loops in supersymmetric gauge theories, we will study their properties and how they are related to physical observables, such as the Bremsstrahlung function that controls to the energy radiated by an accelerating particle. We will give various examples of the computation of their expectations values using perturbative and non-perturbative techniques. For SU(N) superconformal QCD we perform a three-loop calculation of the generalized cusp anomalous dimension of the BPS Wilson loop operator using HQET formalism. We obtain an expression which is valid at generic geometric and internal angles and finite gauge group rank N. For equal and opposite angles this expression vanishes, proving that at these points the cusp becomes BPS. From its small angle expansion we derive the corresponding Bremsstrahlung function at three loops, matching the matrix model prediction given in terms of derivatives of the Wilson loop on the ellipsoid.

In this thesis, the emerging field of higher gauge theory will be discussed, particularly in relation to problems arising in M-theory, such as selfdual strings and the so-called (2,0) theory. This thesis will begin with a Nahm-like construction for selfdual strings using loop space, the space of loops on spacetime. This construction maps solutions of the Basu-Harvey equation, the BPS equation arising in the description of multiple M2-branes, to solutions of a selfdual string equation on loop space. Furthermore, all ingredients of the construction reduce to those of the ordinary Nahm construction when compactified on a circle with all loops restricted to those wrapping the circle. The rest of this thesis, however, will not involve loop space. We will see a Nahm-like construction for the case of infinitely many selfdual strings, suspended between two M5-branes. This is possible since the limit taken renders the fields describing the M5-branes abelian. This avoids the problem which the rest of this thesis focuses on: What fields describe multiple M5-branes? The answer is likely to involve higher gauge theory, a categorification of gauge theory which describes the parallel transport of extended objects. Any theories which involves 3-algebras, including current M2-brane models and the Lambert-Papageorgakis M5-brane model, are examples of higher gauge theories. Recently, a class of models with N = (1, 0) supersymmetry have been found, with significant overlap with algebraic structures in higher gauge theory. This overlap suggests that the full N = (2, 0) theory could involve semistrict L∞-algebras. Finally, we will see some explicit selfdual string solutions, which may fit into these frameworks.

This thesis studies N=(2,2) gauged linear sigma models (GLSMs) and three-dimensional N=2 Chern-Simons-matter theories and their mathematical applications. After a brief review of GLSMs, we systematically study nonabelian GLSMs for symplectic and orthogonal Grassmannians, following up a proposal in the math community. As consistency checks, we have compared global symmetries, Witten indices, and Calabi-Yau conditions to geometric expectations. We also compute their nonabelian mirrors following the recently developed nonabelian mirror symmetry. In addition, for symplectic Grassmannians, we use the effective twisted superpotential on the Coulomb branch of the GLSM to calculate the ordinary and equivariant quantum cohomology of the space, matching results in the math literature. Then we discuss 3d gauge theories with Chern-Simons terms. We propose a complementary method to derive the quantum K-theory relations of projective spaces and Grassmannians from the corresponding 3d gauge theory with a suitable choice of the Chern-Simons levels. In the derivation, we compare to standard presentations in terms of Schubert cycles, and also propose a new description in terms of shifted Wilson lines, which can be generalized to symplectic Grassmannians. Using this method, we are able to obtain quantum K-theory relations, which match known math results, as well as make predictions.
Doctor of Philosophy
In this thesis, we study two specific models of supersymmetric gauge theories, namely two-dimensional N=(2,2) gauged linear sigma models (GLSMs) and three-dimensional N=2 Chern-Simons-matter theories. These models have played an important role in quantum field theory and string theory for decades, and generated many fruitful results, improving our understanding of Nature by drawing on many branches in mathematics, such as complex differential geometry, intersection theory, quantum cohomology/quantum K-theory, enumerative geometry, and many others. Specifically, this thesis is devoted to studying their applications in quantum cohomology and quantum K-theory. In the first part of this thesis, we systematically study two-dimensional GLSMs for symplectic and orthogonal Grassmannians, generalizing the study for ordinary Grassmannians. By analyzing the Coulomb vacua structure of the GLSMs for symplectic Grassmannians, we are able to obtain the ordinary and equivariant quantum cohomology for these spaces. A similar methodology applies to 3d Chern-Simons-matter theories and quantum K-theory, for which we propose a new description in terms of shifted Wilson lines.

We calculate the string tensions, mass spectrum, and deconfining temperatures of SO(N) gauge theories in 2+1 dimensions. After a review of lattice field theory, we describe how we simulate the corresponding lattice gauge theories, construct operators to project on to specific states, and extrapolate values to the continuum limit. We discuss how to avoid possible complications such as finite size corrections and the bulk transition. SO(N) gauge theories have become recently topical since they do not have a fermion sign problem, are orbifold equivalent to SU(N) gauge theories, and share a common large-N limit in their common sector of states with SU(N) gauge theories. This motivates us to compare the physical properties of SO(N) and SU(N) gauge theories between 'group equivalences', which includes Lie algebra equivalences such as SO(6) and SU(4), and particularly a large-N equivalence. We discuss the large-N orbifold equivalence between SO(N) and SU(N) gauge theories, which relates the large-N gauge theories perturbatively. Using large-N extrapolations at fixed 't Hooft coupling, we test to see if SO(N) gauge theories and SU(N) gauge theories share non-perturbative properties at the large-N limit. If these group equivalences lead to similar physics in the gauge theories, then we could imagine doing finite chemical potential calculations that are currently intractable in SU(N) gauge theories by calculating equivalent quantities in the corresponding SO(N) gauge theories. We show that the SO(N) and SU(N) values match between group equivalences and at the large-N limit.

Gauge theories and their BRST invariance are reviewed. Gauge-invariant (color) subamplitudes for non-Abelian gauge theories are discussed. BRST transformations of non-Abelian vertices are derived, and are used to obtain the gauge transformation of any Feynman diagram. From this minimal set of gauge-invariant subamplitudes in perturbation theory can be found. This knowledge is useful in the application of the spinor helicity technique, and is indispensible for future developments of non-Abelian perturbation theories.